# Convolutional unitary or orthogonal recurrent neural networks

arXiv: https://arxiv.org/abs/2302.07396 · HTML: https://mlsanity.com/paper/2302.07396/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 14 Feb 2023.

## Builds on (most cited first, 5 of 5)

- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [A Simple Way to Initialize Recurrent Networks of Rectified Linear Units](https://mlsanity.com/paper/1504.00941/) (2015)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)

## Built on by (most cited first, 2 of 2)

- [On the dynamics of convolutional recurrent neural networks near their critical point](https://mlsanity.com/paper/2405.13854/) (2024)
- [Input-driven circuit reconfiguration in critical recurrent neural networks](https://mlsanity.com/paper/2405.15036/) (2024)

## Abstract (arXiv)

Recurrent neural networks are extremely powerful yet hard to train. One of their issues is the vanishing gradient problem, whereby propagation of training signals may be exponentially attenuated, freezing training. Use of orthogonal or unitary matrices, whose powers neither explode nor decay, has been proposed to mitigate this issue, but their computational expense has hindered their use. Here we show that in the specific case of convolutional RNNs, we can define a convolutional exponential and that this operation transforms antisymmetric or anti-Hermitian convolution kernels into orthogonal or unitary convolution kernels. We explicitly derive FFT-based algorithms to compute the kernels and their derivatives. The computational complexity of parametrizing this subspace of orthogonal transformations is thus the same as the networks' iteration.